Q: a best critical region of two parameters in normal distribution This question is from Introduction to Mathematical Statistics written by Hogg et al
In page 448,
1.6 Let $ X_1, X_2, \ldots, X_{10} $ is a random sample from a distribution that is $N(\theta_1, \theta_2)$. Find a best test of the simple hypothesis $H_0 : \theta _1= \theta'_1 = 0, \theta_2=\theta'_2=1$ against the alternative simple hypothesis $H_1 : \theta _1= \theta''_1=1, \theta_2=\theta''_2=4 $
I briefly write a best critical $C=\lbrace(x_1,x_2, \ldots, x_{10})| 3\sum_{i=1}^{10}x_i^2 +2\sum_{i=1}^{10}x_i \ge k\rbrace$
But we have to consider a joint distribution of chi square and normal. Is that right? That looks so complicated so I feel unsure of a above expression. Help me please!

A: You can manipulate your expression finding the following critical region
$$\Sigma_i \left(X_i+\frac{1}{3}\right)^2 \geq k^*$$
Now, under $H_0$, the distribution is known: it's a noncentral chi-squared and you can calculate $k^*$
